Problem: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 7x - 4$, and $ m \angle ABC = 2x + 13$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Answer: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 13} + {7x - 4} = {90}$ Combine like terms: $ 9x + 9 = 90$ Subtract $9$ from both sides: $ 9x = 81$ Divide both sides by $9$ to find $x$ $ x = 9$ Substitute $9$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 7({9}) - 4$ Simplify: $ {m\angle CBD = 63 - 4}$ So ${m\angle CBD = 59}$.